Geometry: Properties of Hexagons

Definition of a Hexagon

• A hexagon is a two-dimensional shape with six sides and six interior angles.
• The term "hexagon" comes from Greek roots "hex" meaning "six" and "gonia" meaning "corner".

Features of a Hexagon

• A hexagon is a polygon, a two-dimensional, closed geometrical shape made up of straight line segments.
• It has six equal sides and angles, making it a regular polygon.

Regular Hexagons

• A regular hexagon has six equal line segments making six equal angles.
• All sides and angles of a regular hexagon are equal in dimension.
• The perimeter of a regular hexagon is the sum of all its sides.
• Each interior angle of a regular hexagon measures 120 degrees.
• A regular hexagon is equiangular, meaning all angles have equal measure.

Angles of a Regular Hexagon

• A regular hexagon can be thought of as consisting of six identical equilateral triangles.
• Each angle of a regular hexagon has two angles of an equilateral triangle, each measuring 60 degrees.

Formula for the Area of a Regular Hexagon

• The formula for the area of a regular hexagon is (3 * sqrt(3) * a^2) / 2, where a is the side length of the hexagon.
• Another formula for the area of a regular hexagon is (3 * (a^2 * sqrt(3)) / 4), where a is the side of the hexagon and d is the height of the hexagon from one of the bases.

Irregular Hexagons

• An irregular hexagon is one that does not have six equal sides and six equal angles.
• Not all sides and angles of an irregular hexagon have to have different measures.

Features of Irregular Hexagons

• All interior angles of an irregular hexagon do not measure 120 degrees.
• All exterior angles of an irregular hexagon do not measure 60 degrees.
• The perimeter of an irregular hexagon is not given by a simple formula, but rather by adding individual side lengths.

Convex Irregular Hexagons

• A convex hexagon is a hexagon with an outward curved design, where all interior angles are less than 180 degrees.

Concave Irregular Hexagons

• A concave hexagon is a hexagon with one or more surfaces curved inwards, where at least one interior angle is greater than 180 degrees.

Complex Hexagons

• Complex hexagons are six-sided, closed structures formed by the intersection of different shapes, such as the Star of David.

Hexagons in Nature

• Hexagons exist naturally in the formation of snowflakes, the crystalline structure of graphene, and in the honeycombs made by bees.
• Hexagonal rocks can sometimes occur naturally, and the center scutes on a turtle's carapace are also hexagonal in shape.